Journal of Mechanical Science and Technology

, Volume 29, Issue 7, pp 2653–2661 | Cite as

A comparison of DAE integrators in the context of benchmark problems for flexible multibody dynamics

  • Peter Betsch
  • Christian Becker
  • Marlon Franke
  • Yinping Yang
  • Alexander Janz


In the present work a uniform framework for general flexible multibody dynamics is used to compare state-of-the-art DAE integrators in the context of benchmark problems. The multibody systems considered herein are comprised of rigid bodies, nonlinear beams and shells. The constitutive laws applied in the benchmark problems belong to the class of hyperelastic materials. To numerically integrate the uniform set of DAEs three alternative time-stepping schemes are applied: (i) an energy-momentum consistent method, (ii) a specific variational integrator and (iii) a generalized-a scheme.


Differential-algebraic equations Multibody systems Nonlinear structural mechanics Structure-preserving discretization Time-stepping schemes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. B. Rubin, Cosserat theories: Shells, rods and points, Solid Mechanics and its Applications, Kluwer Academic Publishers, 79 (2000).Google Scholar
  2. [2]
    P. Betsch and N. Sänger, On the consistent formulation of torques in a rotationless framework for multibody dynamics, Computers & Structures, 127 (2013) 29–38.CrossRefGoogle Scholar
  3. [3]
    P. Betsch and N. Sänger, A nonlinear finite element framework for flexible multibody dynamics: Rotationless formulation and energy-momentum conserving discretization, In Carlo L. Bottasso, editor, Multibody Dynamics: Computational Methods and Applications, Computational Methods in Applied Sciences, Springer-Verlag, 12 (2009) 119–141.Google Scholar
  4. [4]
    D. Negrut, R. Rampalli, G. Ottarsson and A. Sajdak, On an implementation of the Hilber-Hughes-Taylor method in the context of index 3 differential-algebraic equations of multibody dynamics (DETC2005-85096), J. Comput. Nonlinear Dynam., 2 (2007) 73–85.CrossRefGoogle Scholar
  5. [5]
    M. Arnold and O. Brüls, Convergence of the generalized-a scheme for constrained mechanical systems, Multibody System Dynamics, 18 (2) 2007) 185–202.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    S. Leyendecker, J. E. Marsden and M. Ortiz, Variational integrators for constrained dynamical systems, Z. Angew, Math. Mech. (ZAMM), 88 (9) 2008) 677–708.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S. Leyendecker, P. Betsch and P. Steinmann, The discrete null space method for the energy consistent integration of constrained mechanical systems, Part III: Flexible multibody dynamics, Multibody System Dynamics, 19 (1-2) (2008) 45–72.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    M. González, D. Dopico, U. Lugrís and J. Cuadrado, A benchmarking system for MBS simulation software: Problem standardization and performance measurement, Multibody System Dynamics, 16 (2) 2006) 179–190.CrossRefMATHGoogle Scholar
  9. [9]
    M. Valášek, Z. Šika and T. Vampola, Criteria of benchmark selection for efficient flexible multibody system formalisms, Applied and Computational Mechanics, 1 (2007) 351–356.Google Scholar
  10. [10] html.Google Scholar
  11. [11]
    P. Betsch and N. Sänger, On the use of geometrically exact shells in a conserving framework for flexible multibody dynamics, Comput. Methods Appl. Mech. Engrg., 198 (2009) 1609–1630.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    P. Betsch, C. Hesch, N. Sänger and S. Uhlar, Variational integrators and energy-momentum schemes for flexible multibody dynamics, J. Comput. Nonlinear Dynam., 5 (3) (2010) 031001/1-11.Google Scholar
  13. [13]
    A. Cardona and M. Géradin, Time integration of the equations of motion in mechanism analysis, Computers & Structures, 33 (3) 1989) 801–820.CrossRefMATHGoogle Scholar
  14. [14]
    A. Cardona and M. Géradin, Numerical integration of second order differential-algebraic systems in flexible mechanism dynamics, aditted by. S. Pereira and J.A.C. Ambrósio, Computer-Aided Analysis of Rigid and Flexible Mechanical Systems, NATO-ASI Series E: Applied Sciences, Kluwer Academic Publishers, 268 (1994) 501–529.CrossRefGoogle Scholar
  15. [15]
    J. Yen, L. Petzold and S. Raha, A time integration algorithm for flexible mechanism dynamics: The DAE a-method, Comput. Methods Appl. Mech. Engrg., 158 (3-4) (1998) 341–355.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    C. Lunk and B. Simeon, Solving constrained mechanical systems by the family of Newmark and a-methods, Z. Angew, Math. Mech. (ZAMM), 86 (10) 2006) 772–784.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    L. O. Jay and D. Negrut, Extensions of the HHT-a method to differential-algebraic equations in mechanics, Electronic Transactions on Numerical Analysis, 26 (2007) 190–208.MathSciNetMATHGoogle Scholar
  18. [18]
    D. Negrut, L. O. Jay and N. Khude, A discussion of loworder numerical integration formulas for rigid and flexible multibody dynamics, Journal of Computational and Nonlinear Dynamics, 4 (2) (2009) 021008/1-11.Google Scholar
  19. [19]
    O. Brüls and A. Cardona, On the use of Lie group time integrators in multibody dynamics, J. Comput. Nonlinear Dynam., 5 (3) (2010) 031002/1-031002/13.Google Scholar
  20. [20]
    O. Brüls, A. Cardona and M. Arnold, Lie group generalized- a time integration of constrained flexible multibody systems, Mechanism and Machine Theory, 48 (1) 2012) 121–137.CrossRefGoogle Scholar
  21. [21]
    O. A. Bauchau and C. L. Bottass, On the design of energy preserving and decaying schemes for flexible, nonlinear multi-body systems, Comput. Methods Appl. Mech. Engrg., 169 (1-2) (1999) 61–799.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    O. A. Bauchau, Flexible multibody dynamics, Solid Mechanics and its Applications, Springer-Verlag, 176 (2011).Google Scholar
  23. [23]
    J. von Schwerin, Multibody System SIMulation, Springer- Verlag (1999).Google Scholar
  24. [24]
    W. O. Schiehlen, Editor, Multibody systems handbook, Springer-Verlag (1990).Google Scholar
  25. [25]
    E. Hairer and G. Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, Springer-Verlag (1991).Google Scholar
  26. [26]
    N. Khude and D. Negrut, A MATLAB implementation of the seven-body mechanism for implicit integration of the constrained equations of motion, Technical Report TR-2007- 07, Simulation-Based Engineering Laboratory, The University of Wisconsin-Madison (2007).Google Scholar
  27. [27]
    O. A. Bauchau, G. Wu, P. Betsch, A. Cardona, J. Gerstmayr, B. Jonker, P. Masarati and V. Sonneville, Validation of flexible multibody dynamics beam formulations using benchmark problems, Proceedings of the 3rd Joint International Conference on Multibody System Dynamics and the 7th Asian Conference on Multibody Dynamics (IMSDACMD), Busan, Korea, 30 June 30- 3 July (2014).Google Scholar
  28. [28]
    O. A. Bauchau, J.-Y. Choi and C. L. Bottasso, On the modeling of shells in multibody dynamics, Multibody System Dynamics, 8 (2002) 459–489.CrossRefMATHGoogle Scholar
  29. [29]
    O. A. Bauchau, C. L. Bottasso and L. Trainelli, Robust integration schemes for flexible multibody systems, Comput. Methods Appl. Mech. Engrg., 192 (3-4) (2003) 395–420.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    P. Betsch and P. Steinmann, A DAE approach to flexible multibody dynamics, Multibody System Dynamics, 8 (2002) 367–391.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Korean Society of Mechanical Engineers and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Peter Betsch
    • 1
  • Christian Becker
    • 1
  • Marlon Franke
    • 1
  • Yinping Yang
    • 1
  • Alexander Janz
    • 1
  1. 1.Institute of MechanicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations